# two orthogonal vectors that are not linearly indepdent

I need to give an example of two orthogonal vectors that are not linearly independent.

My Workings:

I know that linear dependency of vectors $u$ and $v$ means that there exists number $a$ and $b$, which both cant be 0), such that $au+bv = 0$. So $0=(au+bv)^T(au+bv) = a^2||u||^2+b||v||^2$ because they are orthogonal. SO $A\ne = 0$, then $u=0$. Thus, one of the vectors must be the zero vector.

Thus, would $(1,1)^T$ and $(0,0)^T$ be an example of two orthogonal vectors that are not linearly independent?

• I see a zero vector? That is never lin. independent with any vector. Normally, I am not aware of two orthogonal nonzero vectors that are linear dependent. – imranfat Nov 24 '16 at 3:46

## 1 Answer

Yep! The vector 0 is orthogonal to all vectors and linearly dependent with all vectors.

• is (1,1) and (0,0) are two orthogonal vectors that are not linearly independent? – jh123 Nov 24 '16 at 4:02
• Yep. $\{(0,0), (a,b)\}$ is linearly dependent and orthogonal for any $a,b$. – mrob Nov 24 '16 at 4:11